\newproblem{lay:1_8_30}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.8.30}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Suppose vectors $\mathbf{v}_1$, $\mathbf{v}_2$, ..., $\mathbf{v}_p$ span $\mathbb{R}^n$, and let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear
	transformation. Suppose $T(\mathbf{v}_i)=\mathbf{0}$ for $i=1,2,...,p$. Show that $T$ is the zero transformation. That is, show that if $\mathbf{x}$ is 
	any vector in $\mathbb{R}^n$, then $T(\mathbf{x})=\mathbf{0}$
}{
  % Solution
	If $\mathbf{v}_1$, $\mathbf{v}_2$, ..., $\mathbf{v}_p$ span $\mathbb{R}^n$, then any vector $\mathbf{x}\in\mathbb{R}^n$ can be expressed as a linear combination of
	$\mathbf{v}_i$'s:
	\begin{center}
		$\mathbf{x}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+...+c_p\mathbf{v}_p$
	\end{center}
	Applying the transformation $T$ to $\mathbf{x}$ we get
	\begin{center}
		$\begin{array}{rcll}
		   T(\mathbf{x})&=&T(c_1\mathbf{v}_1+c_2\mathbf{v}_2+...+c_p\mathbf{v}_p) & \text{By definition of }\mathbf{x} \\
			              &=&c_1T(\mathbf{v}_1)+c_2T(\mathbf{v}_2)+...+c_pT(\mathbf{v}_p) & \text{By linearity of }T \\
			              &=&c_1\mathbf{0}+c_2\mathbf{0}+...+c_p\mathbf{0} & \text{As stated by the problem} \\
			              &=&\mathbf{0} & \\
		\end{array}$
	\end{center}
}
\useproblem{lay:1_8_30}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
